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We apply upper and lower compensated convex transforms, which are `tight'one-sided approximations of a given function, to the extraction of finegeometric singularities from semiconvex/semiconcave functions and DC-functionsin $\mathbb{R}^n$ (difference of convex functions). Well-known examples of(locally) semiconcave functions include the Euclidean distance and squareddistance functions. For a locally semiconvex function $f$ with general modulus,we show that `locally' a point is a singular (non-differentiable) point if andonly if it is a scale $1$-valley point, and if $x$ is a singular point, thenlocally the limit of the scaled valley transform exists at every point $x$ and$ \lim_{\lambda\to \infty}\lambda V_\lambda (f)(x)=r_x^2/4$, where $r_x$ is theradius of the minimal bounding sphere of the (Fr\'echet) subdifferential$\partial_- f(x)$ and $V_\lambda (f)(x)$ is the valley transform at $x$. Thusthe limit function $\mathcal{V}_\infty(f)(x):=\lim_{\lambda\to+\infty}\lambdaV_\lambda (f)(x)=r_x^2/4$ gives a `scale $1$-valley landscape function' of thesingular set for a locally semiconvex function $f$, and also provides anasymptotic expansion of the upper transform $C^u_\lambda(f)(x)$ when $\lambda\to \infty$. For a locally semiconvex function $f$ with linear modulus we showthat the limit of the gradient of the upper compensated convex transform$\lim_{\lambda\to+\infty}\nabla C^u_\lambda(f)(x)$ exists and equals the centreof the minimal bounding sphere of $\partial_- f(x$, and that for a DC-function$f=g-h$, the scale $1$-edge transform satisfies$\liminf_{\lambda\to+\infty}\lambda E_\lambda (f)(x)\geq(r_{g,x}-r_{h,x})^2/4$, where $r_{g,x}$ and $r_{h,x}$ are the radii of theminimal bounding spheres of the subdifferentials $\partial_- g$ and $\partial_-h$ of the convex functions $g$ and $h$ at $x$ respectively.Comment: A Chinese version of the material in this manuscript has been  published in Zhang, Kewei, Crooks, Elaine and Orlando, Antonio, Compensated  convex transforms and geometric singularity extraction from semiconvex  functions (in Chinese), Sci. Sin. Math., 46 (2016) 747-768, doi:  10.1360/N012015-0033

Compensated convex transforms and geometric singularity\\ extraction from semiconvex functions